Gottfried Wilhelm (von) Leibniz (sometimes spelled Leibnitz) (/ˈlaɪbnɪts/;^[11] German: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts]^[12][13] or [ˈlaɪpnɪts];^[14] French: Godefroi Guillaume Leibnitz;^[15] 1 July 1646 [O.S. 21 June] – 14 November 1716) was a prominent German polymath and one of the most important logicians, mathematicians and natural philosophers of the Enlightenment. As a representative of the seventeenth-century tradition of rationalism, Leibniz's most prominent accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments.^[16] Mathematical works have consistently favored Leibniz's notation as the conventional expression of calculus. It was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation (by means of non-standard analysis). He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685^[17] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of all digital computers.

In philosophy, Leibniz is most noted for his optimism, i.e. his conclusion that our universe is, in a restricted sense, the best possible one that God could have created, an idea that was often lampooned by others such as Voltaire. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also assimilates elements of the scholastic tradition, notably that conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence.

Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibniz also contributed to the field of library science. While serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would serve as a guide for many of Europe's largest libraries.^[18] Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. He wrote in several languages, but primarily in Latin, French, and German.^[19] There is no complete gathering of the writings of Leibniz translated into English.^[20]

Gottfried Leibniz was born on 1 July 1646, toward the end of the Thirty Years' War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his family journal:

In English:

Leibniz was baptized on 3 July of that year at St. Nicholas Church, Leipzig; his godfather was the Lutheran theologian Martin Geier.^[23] His father died when he was six years old, and from that point on he was raised by his mother.^[24]

Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his father's personal library. He was given free access to it from the age of seven. While Leibniz's schoolwork was largely confined to the study of a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years.^[25] Access to his father's library, largely written in Latin, also led to his proficiency in the Latin language, which he achieved by the age of 12. He also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13.^[26]

In April 1661 he enrolled in his father's former university at age 14,^[27][1][28] and completed his bachelor's degree in Philosophy in December 1662. He defended his Disputatio Metaphysica de Principio Individui (Metaphysical Disputation on the Principle of Individuation),^[29] which addressed the principle of individuation, on 9 June 1663. Leibniz earned his master's degree in Philosophy on 7 February 1664. He published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum (An Essay of Collected Philosophical Problems of Right),^[29] arguing for both a theoretical and a pedagogical relationship between philosophy and law, in December 1664. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665.^[30] His dissertation was titled De conditionibus (On Conditions).^[29]

In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria (On the Combinatorial Art), the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666.^[29][31] His next goal was to earn his license and Doctorate in Law, which normally required three years of study. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth.^[32][33] Leibniz subsequently left Leipzig.^[34]

Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, which he had probably been working on earlier in Leipzig.^[35] The title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure (Inaugural Disputation on Ambiguous Legal Cases).^[29] Leibniz earned his license to practice law and his Doctorate in Law in November 1666. He next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction".^[36]

As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Many posthumously published editions of his writings presented his name on the title page as "Freiherr G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility.^[37]

Leibniz's first position was as a salaried secretary to an alchemical society in Nuremberg.^[38] He knew fairly little about the subject at that time but presented himself as deeply learned. He soon met Johann Christian von Boyneburg (1622–1672), the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn.^[39] Von Boyneburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for the Electorate.^[40] In 1669, Leibniz was appointed assessor in the Court of Appeal. Although von Boyneburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.

Von Boyneburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. After Leibniz's service to the Elector there soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main force in European geopolitics during Leibniz's adult life was the ambition of Louis XIV of France, backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion,^[41] but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting, late implementation of Leibniz's plan, after the Eastern hemisphere colonial supremacy in Europe had already passed from the Dutch to the British.

Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus. He met Nicolas Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published.^[42] He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives.

When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673.^[43] There Leibniz came into acquaintance of Henry Oldenburg and John Collins. He met with the Royal Society where he demonstrated a calculating machine that he had designed and had been building since 1670. The machine was able to execute all four basic operations (adding, subtracting, multiplying, and dividing), and the society quickly made him an external member.

The mission ended abruptly when news of the Elector's death (12 February 1673) reached them. Leibniz promptly returned to Paris and not, as had been planned, to Mainz.^[44] The sudden deaths of his two patrons in the same winter meant that Leibniz had to find a new basis for his career.

In this regard, a 1669 invitation from Duke John Frederick of Brunswick to visit Hanover proved to have been fateful. Leibniz had declined the invitation, but had begun corresponding with the duke in 1671. In 1673, the duke offered Leibniz the post of counsellor. Leibniz very reluctantly accepted the position two years later, only after it became clear that no employment was forthcoming in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court.^[45]

In 1675 he tried to get admitted to the French Academy of Sciences as a foreign honorary member, but it was considered that there were already enough foreigners there and so no invitation came. He left Paris in October 1676.

Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where Newton accused him of having seen Newton's unpublished work on calculus in advance.^[46] This was alleged to be evidence supporting the accusation, made decades later, that he had stolen calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met van Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who had just completed his masterwork, the Ethics.^[47]

In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.

Leibniz began promoting a project to use windmills to improve the mining operations in the Harz Mountains. This project did little to improve mining operations and was shut down by Duke Ernst August in 1685.^[45]

Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain.^[48]

The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire. The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of England, once both King William III and his sister-in-law and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament.

The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.

The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.

Leibniz was appointed Librarian of the Herzog August Library in Wolfenbüttel, Lower Saxony, in 1691.

In 1708, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarised Newton's calculus.^[49] Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of calculus.

In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hanover and met Leibniz, who then took some interest in Russian matters for the rest of his life. In 1712, Leibniz began a two-year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714.

Leibniz died in Hanover in 1716. At the time, he was so out of favor that neither George I (who happened to be near Hanover at that time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honor his death. His grave went unmarked for more than 50 years. Leibniz was eulogized by Fontenelle, before the French Academy of Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.

Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy.

On the other hand, he was charming, well-mannered, and not without humor and imagination.^[50] He had many friends and admirers all over Europe. Though he identified as a protestant, Leibniz learned to appreciate aspects of Catholicism through his patrons and colleagues. He never admitted the Protestant view of the Pope as an Antichrist.^[51] Leibniz was claimed as a philosophical theist.^{[52][53][54][55]} Leibniz remained committed to Trinitarian Christianity throughout his life.^[56]

Leibniz's philosophical thinking appears fragmented, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He wrote only two book-length philosophical treatises, of which only the Théodicée of 1710 was published in his lifetime.

Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld;^[57] it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances".^[58] Between 1695 and 1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms.

Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions,^[59] especially when these were inconsistent with Christian orthodoxy.

Unlike Descartes and Spinoza, Leibniz had a thorough university education in philosophy. He was influenced by his Leipzig professor Jakob Thomasius, who also supervised his BA thesis in philosophy.^[9] Leibniz also eagerly read Francisco Suárez, a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed their work through a lens heavily tinted by scholastic notions. Yet it remains the case that Leibniz's methods and concerns often anticipate the logic, and analytic and linguistic philosophy of the 20th century.

Leibniz variously invoked one or another of seven fundamental philosophical Principles:^[60]

Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted.^[66]

Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. He proposes his theory that the universe is made of an infinite number of simple substances known as monads.^[67] Monads can also be compared to the corpuscles of the Mechanical Philosophy of René Descartes and others. These simple substances or monads are the "ultimate units of existence in nature". Monads have no parts but still exist by the qualities that they have. These qualities are continuously changing over time, and each monad is unique. They are also not affected by time and are subject to only creation and annihilation.^[68] Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal.

Leibniz's proof of God can be summarized in the Théodicée.^[69] Reason is governed by the principle of contradiction and the principle of sufficient reason. Using the principle of reasoning, Leibniz concluded that the first reason of all things is God.^[69] All that we see and experience is subject to change, and the fact that this world is contingent can be explained by the possibility of the world being arranged differently in space and time. The contingent world must have some necessary reason for its existence. Leibniz uses a geometry book as an example to explain his reasoning. If this book was copied from an infinite chain of copies, there must be some reason for the content of the book.^[70] Leibniz concluded that there must be the "monas monadum" or God.

The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of pre-established harmony, each monad follows a pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic.

Monads are purported to have gotten rid of the problematic:

The Theodicy^[71] tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws.

Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity.^[72] This project was motivated in part by Leibniz's belief, shared by many conservative philosophers and theologians during the Enlightenment, in the rational and enlightened nature of the Christian religion as compared to its purportedly less advanced non-Western counterparts. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science).

Because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism:^[73] if God is all good, all wise and all powerful, then how did evil come into the world? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions and ineffective actions in the exercise of their free will. God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin) and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good.^[74]

Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination.

For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome (VI). Along these lines, he declares that every type of perfection "pertains to him (God) in the highest degree" (I). Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly (III). Because God cannot act imperfectly, the decisions he makes pertaining to the world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured. However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition (IV). In accordance with this, many act as rebels, but Leibniz says that the only way we can truly love God is by being content "with all that comes to us according to his will" (IV).

Because God is "an absolutely perfect being" (I), Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of (III). His syllogism then ends with the statement that God has made the world perfectly in all ways. This also affects how we should view God and his will. Leibniz states that, in lieu of God’s will, we have to understand that God "is the best of all masters" and he will know when his good succeeds, so we, therefore, must act in conformity to his good will—or as much of it as we understand (IV). In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so. Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition (II). Leibniz then asserts that different principles and geometry cannot simply be from the will of God, but must follow from his understanding.^[75]

Leibniz wrote: "Why is there something rather than nothing? The sufficient reason ... is found in a substance which ... is a necessary being bearing the reason for its existence within itself."^[76] Martin Heidegger called this question "the fundamental question of metaphysics".^[77][78]

Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:

Leibniz's calculus ratiocinator, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda^[80] that can now be read as groping attempts to get symbolic logic—and thus his calculus—off the ground. These writings remained unpublished until the appearance of a selection edited by C.I. Gerhardt (1859). L. Couturat published a selection in 1901; by this time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege.

Leibniz thought symbols were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this. His notation for calculus is an example of his skill in this regard. Peirce, a 19th-century pioneer of semiotics, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a well-running logic and mathematics.

But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry, he deemed not real.^[81] Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character:

Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.

Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established.^[83] Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as Turing completeness, where computation was used to define equivalent universal languages (see Turing degree).

Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege.^[84] Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:

The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse.

Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his History of Western Philosophy, Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.

Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims (e.g., that each of the fundamental monads mirrors the whole universe) follow logically from Leibniz's conscious choice to reject relations between things as unreal. He regarded such relations as (real) qualities of things (Leibniz admitted unary predicates only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of De Morgan, Peirce, Schröder and Russell himself, now standard in predicate logic. Notably, Leibniz also declared space and time to be inherently relational.^[85]

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.^[86] In the 18th century, "function" lost these geometrical associations. Leibniz also believed that the sum of an infinite number of zeros would equal to one half using the analogy of the creation of the world from nothing.^[87] Leibniz was also one of the pioneers in actuarial science, calculating the purchase price of life annuities and the liquidation of a state's debt.^[88]

Leibniz's discoveries of Boolean algebra and of symbolic logic, also relevant to mathematics, are discussed in the preceding section. The best overview of Leibniz's writings on calculus may be found in Bos (1974).^[89]

The Leibniz formula for π states that

Leibniz wrote that circles "can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted".^[97] However this formula is only accurate with a large number of terms, using 10,000,000 terms to obtain the correct value of π/4 to 8 decimal places.^[98] Leibniz attempted to create a definition for a straight line while attempting to prove the parallel postulate.^[99] While most mathematicians defined a straight line as the shortest line between two points, Leibniz believed that this was merely a property of a straight line rather than the definition.^[100]

Leibniz is credited, along with Sir Isaac Newton, with the discovery of calculus (differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = f(x).^[101] He introduced several notations used to this day, for instance the integral sign ∫, representing an elongated S, from the Latin word summa, and the d used for differentials, from the Latin word differentia. Leibniz did not publish anything about his calculus until 1684.^[102] Leibniz expressed the inverse relation of integration and differentiation, later called the fundamental theorem of calculus, by means of a figure^[103] in his 1693 paper Supplementum geometriae dimensoriae....^[104] However, James Gregory is credited for the theorem's discovery in geometric form, Isaac Barrow proved a more generalized geometric version, and Newton developed supporting theory. The concept became more transparent as developed through Leibniz's formalism and new notation.^[105] The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule.

Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and also in De Motu, criticized these. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.^[106]

From 1711 until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented calculus independently of Newton. This subject is treated at length in the article Leibniz–Newton calculus controversy.

The use of infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass,^[107] but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory, in the context of a field of hyperreal numbers. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity, while the standard part function implements the Leibnizian transcendental law of homogeneity.

Leibniz was the first to use the term analysis situs,^[108] later used in the 19th century to refer to what is now known as topology. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal, argues:

But Hideaki Hirano argues differently, quoting Mandelbrot:^[110]

Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: Natura non facit saltus.^[63] We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.

Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings.

Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.^[112]

Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions."^[113]

Leibniz held a relationist notion of space and time, against Newton's substantivalist views.^{[114][115][116]} According to Newton's substantivalism, space and time are entities in their own right, existing independently of things. Leibniz's relationism, on the other hand, describes space and time as systems of relations that exist between objects. The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light.

One of Leibniz's projects was to recast Newton's theory as a vortex theory.^[117] However, his project went beyond vortex theory, since at its heart there was an attempt to explain one of the most difficult problems in physics, that of the origin of the cohesion of matter.^[117]

The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy, a recent direction in cosmology, claim Leibniz as a precursor. In addition to his theories about the nature of reality, Leibniz's contributions to the development of calculus have also had a major impact on physics.

Leibniz's vis viva (Latin for "living force") is mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter.^[118] Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. In reality, both energy and momentum are conserved, so the two approaches are equally valid.

By proposing that the earth has a molten core, he anticipated modern geology. In embryology, he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology, he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea, unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory.^[119] In medicine, he exhorted the physicians of his time—with some results—to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.

Psychology had been a central interest of Leibniz.^[120][121] He appears to be an "underappreciated pioneer of psychology" ^[122] He wrote on topics which are now regarded as fields of psychology: attention and consciousness, memory, learning (association), motivation (the act of "striving"), emergent individuality, the general dynamics of development (evolutionary psychology). His discussions in the New Essays and Monadology often rely on everyday observations such as the behaviour of a dog or the noise of the sea, and he develops intuitive analogies (the synchronous running of clocks or the balance spring of a clock). He also devised postulates and principles that apply to psychology: the continuum of the unnoticed petite perceptions to the distinct, self-aware apperception, and psychophysical parallelism from the point of view of causality and of purpose: “Souls act according to the laws of final causes, through aspirations, ends and means. Bodies act according to the laws of efficient causes, i.e. the laws of motion. And these two realms, that of efficient causes and that of final causes, harmonize with one another.” ^[123] This idea refers to the mind-body problem, stating that the mind and brain do not act upon each other, but act alongside each other separately but in harmony.^[124] Leibniz, however, did not use the term psychologia.^[125] Leibniz’ epistemological position—against John Locke and English empiricism (sensualism)—was made clear: “Nihil est in intellectu quod non fuerit in sensu, nisi intellectu ipse.” – “Nothing is in the intellect that was not first in the senses, except the intellect itself.” ^[126] Principles that are not present in sensory impressions can be recognised in human perception and consciousness: logical inferences, categories of thought, the principle of causality and the principle of purpose (teleology).

Leibniz found his most important interpreter in Wilhelm Wundt, founder of psychology as a discipline. Wundt used the "… nisi intellectu ipse" quotation 1862 on the title page of his Beiträge zur Theorie der Sinneswahrnehmung (Contributions on the Theory of Sensory Perception) and published a detailed and aspiring monograph on Leibniz^[127] Wundt shaped the term apperception, introduced by Leibniz, into an experimental psychologically based apperception psychology that included neuropsychological modelling – an excellent example of how a concept created by a great philosopher could stimulate a psychological research program. One principle in the thinking of Leibniz played a fundamental role: “the principle of equality of separate but corresponding viewpoints.” Wundt characterized this style of thought (perspectivism) in a way that also applied for him—viewpoints that "supplement one another, while also being able to appear as opposites that only resolve themselves when considered more deeply."^[128][129]
Much of Leibniz's work went on to have a great impact on the field of psychology.^[130] Leibniz thought that there are many petites perceptions, or small perceptions of which we perceive but of which we are unaware. He believed that by the principle that phenomena found in nature were continuous by default, it was likely that the transition between conscious and unconscious states had intermediary steps.^[131] For this to be true, there must also be a portion of the mind of which we are unaware at any given time. His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep. In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious. Leibniz was a direct influence on Ernst Platner, who is credited with originally coining the term Unbewußtseyn (unconscious).^[132] Additionally, the idea of subliminal stimuli can be traced back to his theory of small perceptions.^[133] Leibniz's ideas regarding music and tonal perception went on to influence the laboratory studies of Wilhelm Wundt.^[134]

In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. He worked to set up a coherent medical training program, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade. He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.

In 1906, Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxi, he urged that theory be combined with practical application, and thus has been claimed as the father of applied science. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin, he created a steam engine. He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but did not succeed.^[135]

Leibniz may have been the first computer scientist and information theorist.^[136] Early in life, he documented the binary numeral system (base 2), then revisited that system throughout his career.^[137] While Leibniz was examining other cultures to compare his metaphysical views, he encountered an ancient Chinese book I Ching. Leibniz interpreted a diagram which showed yin and yang and corresponded it to a zero and one.^[138] More information can be found in the Sinophile section. Leibniz may have plagiarized Juan Caramuel y Lobkowitz and Thomas Harriot, who independently developed the binary system, as he was familiar with their works on the binary system.^[139] Juan Caramuel y Lobkowitz worked extensively on logarithms including logarithms with base 2.^[140] Thomas Harriot's manuscripts contained a table of binary numbers and their notation, which demonstrated that any number could be written on a base 2 system.^[141] Regardless, Leibniz simplified the binary system and articulated logical properties such as conjunction, disjunction, negation, identity, inclusion, and the empty set.^[142] He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1961, Norbert Wiener suggested that Leibniz should be considered the patron saint of cybernetics.^[143]

In 1671, Leibniz began to invent a machine that could execute all four arithmetic operations, gradually improving it over a number of years. This "stepped reckoner" attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover by a craftsman working under his supervision. They were not an unambiguous success because they did not fully mechanize the carry operation. Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations.^[144] Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher in 2010.^[145] In 1693, Leibniz described a design of a machine which could, in theory, integrate differential equations, which he called "integraph".^[146]

Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards.^[147][148] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.

Later in Leibniz’s career (after the death of von Boinburg), Leibniz moved to Paris and accepted a position as a librarian in the Hanoverian court of Johann Friedrich, Duke of Brunswick-Luneburg.^[149] Leibniz’s predecessor, Tobias Fleischer, had already created a cataloging system for the Duke’s library but it was a clumsy attempt. At this library, Leibniz focused more on advancing the library than on the cataloging. For instance, within a month of taking the new position, he developed a comprehensive plan to expand the library. He was one of the first to consider developing a core collection for a library and felt “that a library for display and ostentation is a luxury and indeed superfluous, but a well-stocked and organized library is important and useful for all areas of human endeavor and is to be regarded on the same level as schools and churches”.^[150] Unfortunately, Leibniz lacked the funds to develop the library in this manner. After working at this library, by the end of 1690 Leibniz was appointed as privy-councilor and librarian of the Bibliotheca Augusta at Wolfenbuettel. It was an extensive library with at least 25,946 printed volumes.^[150] At this library, Leibniz sought to improve the catalog. He was not allowed to make complete changes to the existing closed catalog, but was allowed to improve upon it so he started on that task immediately. He created an alphabetical author catalog and had also created other cataloging methods that were not implemented. While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science. He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University. He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library.

He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis, calculus ratiocinator, and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages (e.g., Esperanto and its rivals), symbolic logic, even the World Wide Web.

Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Académie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg, Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.^[151]

With the possible exception of Marcus Aurelius, no philosopher has ever had as much experience with practical affairs of state as Leibniz. Leibniz's writings on law, ethics, and politics^[152] were long overlooked by English-speaking scholars, but this has changed of late.^[153]

While Leibniz was no apologist for absolute monarchy like Hobbes, or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke, views invoked in support of liberalism, in 18th-century America and later elsewhere. The following excerpt from a 1695 letter to Baron J. C. Boyneburg's son Philipp is very revealing of Leibniz's political sentiments:

In 1677, Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and would be free to vote their consciences;^[155] this is sometimes considered an anticipation of the European Union. He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.

But at the same time, he arrived to propose an interreligious and multicultural project to create a universal system of justice, which required from him a broad interdisciplinary perspective. In order to propose it, he combined linguistics (especially sinology), moral and legal philosophy, management, economics, and politics.^[156]

Leibniz devoted considerable intellectual and diplomatic effort to what would now be called ecumenical endeavor, seeking to reconcile first the Roman Catholic and Lutheran churches, and later the Lutheran and Reformed churches. In this respect, he followed the example of his early patrons, Baron von Boyneburg and the Duke John Frederick—both cradle Lutherans who converted to Catholicism as adults—who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others. (The House of Brunswick remained Lutheran, because the Duke's children did not follow their father.) These efforts included corresponding with French bishop Jacques-Bénigne Bossuet, and involved Leibniz in some theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.

Leibniz the philologist was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way. He refuted the belief, widely held by Christian scholars in his day, that Hebrew was the primeval language of the human race. He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto-Swedish was the ancestor of the Germanic languages. He puzzled over the origins of the Slavic languages and was fascinated by classical Chinese. Leibniz was also an expert in the Sanskrit language.^[87]

He published the princeps editio (first modern edition) of the late medieval Chronicon Holtzatiae, a Latin chronicle of the County of Holstein.

Leibniz was perhaps the first major European intellectual to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China. Having read Confucius Sinarum Philosophus on the first year of its publication,^[158] he concluded that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 000000 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.^[159] Leibniz communicated his ideas of the binary system representing Christianity to the Emperor of China, hoping it would convert him.^[87] Leibniz may be the only major Western philosopher who has attempted to accommodate Confucian ideas to prevailing European beliefs.^[160]

Leibniz's attraction to Chinese philosophy originates from his perception that Chinese philosophy was similar to his own.^[158] The historian E.R. Hughes suggests that Leibniz's ideas of "simple substance" and "pre-established harmony" were directly influenced by Confucianism, pointing to the fact that they were conceived during the period when he was reading Confucius Sinarum Philosophus.^[158]

While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna between May 1688 and February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains. His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican, and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics.

Leibniz also wrote a short paper, Primae veritates, first published by Louis Couturat in 1903 (pp. 518–523)^[161] summarizing his views on metaphysics. The paper is undated; that he wrote it while in Vienna in 1689 was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–90.^[162] Couturat's reading of this paper was the launching point for much 20th-century thinking about Leibniz, especially among analytic philosophers. But after a meticulous study of all of Leibniz's philosophical writings up to 1688—a study the 1999 additions to the critical edition made possible—Mercer (2001) begged to differ with Couturat's reading; the jury is still out.

When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée,^[163] whose supposed central argument Voltaire lampooned in his popular book Candide, which concludes with the character Candide saying, "Non liquet" (it is not clear), a term that was applied during the Roman Republic to a legal verdict of "not proven". Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description. Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. He also influenced David Hume, who read his Théodicée and used some of his ideas.^[164] In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent proponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.

Much of Europe came to doubt that Leibniz had discovered calculus independently of Newton, and hence his whole work in mathematics and physics was neglected. Voltaire, an admirer of Newton, also wrote Candide at least in part to discredit Leibniz's claim to having discovered calculus and Leibniz's charge that Newton's theory of universal gravitation was incorrect.

Leibniz's long march to his present glory began with the 1765 publication of the Nouveaux Essais, which Kant read closely. In 1768, Louis Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by a number of editions, including those edited by Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp, and Mollat. Publication of Leibniz's correspondence with notables such as Antoine Arnauld, Samuel Clarke, Sophia of Hanover, and her daughter Sophia Charlotte of Hanover, began.

In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics.^[165] Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. They made Leibniz somewhat respectable among 20th-century analytical and linguistic philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann). For example, Leibniz's phrase salva veritate, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary literature on Leibniz did not really blossom until after World War II. This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1904–1985) through his translations and his interpretive essays in LeClerc (1973).

Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive.^[166] Analytic and contemporary philosophy continue to invoke his notions of identity, individuation, and possible worlds. Work in the history of 17th- and 18th-century ideas has revealed more clearly the 17th-century "Intellectual Revolution" that preceded the better-known Industrial and commercial revolutions of the 18th and 19th centuries.

In 1985, the German government created the Leibniz Prize, offering an annual award of 1.55 million euros for experimental results and 770,000 euros for theoretical ones. It was the worlds largest prize for scientific achievement prior to the Fundamental Physics Prize.

The collection of manuscript papers of Leibniz at the Gottfried Wilhelm Leibniz Bibliothek – Niedersächische Landesbibliothek was inscribed on UNESCO's Memory of the World Register in 2007.^[167]

Leibniz mainly wrote in three languages: scholastic Latin, French and German. During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg, most notably the "De jure suprematum" a major consideration of the nature of sovereignty.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogue of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1700, remains unpublished, and much of what is published has appeared only in recent decades. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described in a letter as follows:

The extant parts of the critical edition^[169] of Leibniz's writings are organized as follows:

The systematic cataloguing of all of Leibniz's Nachlass began in 1901. It was hampered by two world wars and then by decades of German division into two states with the Cold War's "iron curtain" in between, separating scholars, and also scattering portions of his literary estates. The ambitious project has had to deal with writings in seven languages, contained in some 200,000 written and printed pages. In 1985 it was reorganized and included in a joint program of German federal and state (Länder) academies. Since then the branches in Potsdam, Münster, Hanover and Berlin have jointly published 57 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance works.

The year given is usually that in which the work was completed, not of its eventual publication.

Six important collections of English translations are Wiener (1951), Parkinson (1966), Loemker (1969), Ariew and Garber (1989), Woolhouse and Francks (1998), and Strickland (2006). The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.^[169]

Leibniz is receiving popular attention. The Google Doodle for July 1, 2018 celebrated Leibniz's 372nd birthday.^{[171][172][173]} Using a quill, his hand is shown writing "Google" in binary ASCII code.

One of the earliest popular but indirect expositions of Leibniz was Voltaire's satire Candide, published in 1759. Leibniz was lampooned as Professor Pangloss, described as "the greatest philosopher of the Holy Roman Empire".

Leibniz also appears as one of the main historical figures in Neal Stephenson's series of novels The Baroque Cycle. Stephenson credits readings and discussions concerning Leibniz for inspiring him to write the series.^[174]